Construction of ergodic transformations

k-p4j

Cvm+l':-- ( p + 1)m ' l,

2 <~r<~ m, j = k + p ( m - - 1 ) + r - -

O,

otherwise.

l

Cpm + r,j :

4 k + (p + 1 ) ( m - - 1), 1,

C is clearly contiguous. It remains only to show that zcC = re, and that g(C) is strongly connected and possesses an aperiodic vertex. (i)

For j =

1, we have, ( k - 1)m + 1

X~

~iCi.j --

~.., f=l

n

k

-- m

= ~1.

For 2 ~
k-2

~ici,j : i=1

n

E ~pm+lCprn+l,J ~ k p=o

k-2

1

= p=~-'_: ~ ~pm + 1 (p + 1)m + m

k 2 (p + 1)m x£

o-'r-s (p + 1)m

+m=m+j-

l=rr:.

N o w consider (k+ 1)+s(m-1)~
7[i Ci,j :

\p~--s ~Pm-LlCpm+I'J

-~ TTJ-k+s+lCJ k+s+l,j

= ( k - - s - - 1)+ ( j - - k + s + m ) =m+j--

l=~j.

Hence 7~C = ~. (ii)

We now show that g(C) is strongly connected.

ERGODIC TRANSFORMATIONS

231

(a) We claim that for all i 4: n, there exists a path from vertex i to vertex ( k - 1)m + 1 = I. By examining the construction, it is clear that

V i=/:l,

3j> i~au=

l.

(3.1)

Let f ( i ) be the smallest such j. Then i,f(i),f2(i),...,fk(i) defines a path starting at i and visiting vertices of strictly increasing number. Since there is a finite number of vertices, this sequence must terminate. Moreover, it must terminate at vertex l by virtue of (3.1). (b) Since there are edges from vertex number l to vertices 1, 2 ..... k, the set of vertices V_ 1 = {1 ..... k, I} is strongly connected. By (a) there exists a path from all other vertices into this set (specifically l). Therefore, it suffices to show that there are paths from this set to the remaining vertices. Let Vs={k+s(m-1)+l,k+s(rn-1)+2 ..... k + s ( m - 1 ) + m - 1 } , where 0 ~< s ~< k - 2. N o w there exists an edge from vertex 1 to each node in V0. Therefore, V_ 1 U V0 is strongly connected. We proceed by induction. Suppose V_~ U VoU ... U Vi_l, i < k - 3, is strongly connected. We claim 1,_)}=_1 Vj is strongly connected. The induction step is proved as follows: there are edges from vertex i m + 1 to all the vertices in V i. Therefore, it suffices to show that node i m + 1 E 0}2_~_~ Vj. By the construction, this is true if i m + 1 ~ k + i(m - 1), i.e., if i ~< k - 1, which is certainly true. (iii) It remains to show that A = g ( C ) k > 2, p can take on the value k - 2. Hence

C~k-~),,+ 1# = 1~kin,

2 <~j <~k + (k - 1)(m -- 1).

Therefore, the ( ( k - 2 ) m + 1)st row of k + ( k - 1 ) ( m - 1 ) ) ( k - 2 ) m + 1 for k that a~ > 0. Thus A and C are primitive. ~ - i n v a r i a n t for k > 2. Let C be the (m + 1) X (m + 1) matrix

A intersects the diagonal, since > 2. Hence, there exists an i such This completes the proof that 7r is constructed for k = 2, i.e.,

0

1 rn 0

1 m .......... 1

0

0

0

1

i ~

0 . . . . . . . . . .

0

C =

has a n a p e r i o d i c vertex. For

i

i'

0

0

where the l's are on the superdiagonal. It is clear that the (m + 1)th entries on the diagonals of C 2 and C J are greater than 0. Thus {n>~ 1: C(n)m+l,m+l > 0} ~ {2, 3}, and therefore g.c.d. {n>~l: cm+l,m, r ( n ) -1 > O} is equal to

232

FRIEDMAN

AND

BOYARSKY

1. Hence the (m + 1)st vertex is aperiodic. The graph of g(C) is strongly connected. Therefore, A is primitive and C C ~ . Q.E.D.

Remark. It is interesting to note that for k = n, the matrix C constructed by this result is the skew triangular matrix 1

C=

1 3

/I /

1

1

1

2

2

1 3

1 3

J

//

1

n

n

Clearly, C ~ 3 . On the other hand, the matrix obtained for this same vector, using the construction of Section 2, is

C,

where the l's are on the superdiagonal. Here, C' E ~ . EXAMPLE 3.1.

Let m = 4, n = 12. Thus k = 3, and the vector n = (4, 5, 6, 7, 8, 9, 10, 11, 12)

is ~-invariant, where the associated matrix C is 0

0

I

1

1

1

0

0

0

1

1

1 1 0

1

~

1

~

1

~

1

~

1

~

1

~

~

i

! 1 3

1 3

l 3

0

0

0

0

0

1 0

ERGODIC

233

TRANSFORMATIONS

Theorem 3.1 can be stated in a more general form, but the additional notation might obscure the proof. We, therefore, present the generalization as a corollary. COROLLARY 3.1. Let m , n , j E N be fixed with n = m + l j , ji m , and m = n/k (i.e., k I m + lj'). Then

rc=(m,m+j, rn+ 2j..... n) is ~-invariant. Proof Construct the [ ( k - 1 ) ( m / j ) + 1] square matrix C in a manner directly analogous to that in Theorem 3.1. c(k_,)tm/j,+,,s=lk,

l~s~k,

0,

otherwise.

ForO~p~k-2,

l 1 e(pm/j)+1,, =

k-p<~s~(p+

(P + 1)m ' O,

l) ( j - 1 )

+k,

otherwise

and 1,

if j = k + p ( j - 1 ) + r - 1 ,

O,

otherwise,

C(pm/j)+r's : I where 2 <, r ~ m/j. The proof that Theorem 3.1.

C E~

is analogous to the proof of this fact in Q.E.D.

EXAMPLE 3.2. re = (14, 21, 28, 35, 42, 49, 56). H e r e m = 1 4 , j = 7 , k = 4 a n d l = 6 . The 7 × 7 m a t r i x i s

C=

~-0

0

0

1 2

1 2

0

0

0 1

0 1 _1 !

0

0

0 1

0 0

0

0

4

4

0

0 !

0

0

0

1

0

!

!

1

1

!

6

6

6

6

6

6

0

0

0

0

0

1

1 4

! 4

1 4

0

0

0

0 0 ± ,,,. 4

4

4

234

FRIEDMAN AND BOYARSKY

EXAMPLE 3.3. For m = 10, j---- 2, k = 3, zc = (10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30). The 11 X I1 matrix is

5

3 1

5

3

s

1

C=

1

1

1

1

l

1

1

1

1

l

10

10

10

10

10

10

10

10

10

10

0

0

1 1 1 1

4.

~1

CONCATENATION

~1

~1

0

0

0

0

0

A N D P E R M U T A T I O N OF ~ - I N V A R I A N T

VECTORS

We have seen that it is possible to construct large classes of ~-invariant vectors n. In this section, we shall present results enabling one to construct new c~-invariant vectors from those known to be c~-invariant. The ability to do this is important when one wants to approximate a general ergodic density function f * with a piecewise constant function n. We shall have more to say on this in Section 5. 4.1. C o n c a t e n a t i o n R e s u l t s

THEOREM 4.1. L e t ~ a n d cr be ~ - i n v a r i a n t vectors o f length n 1 a n d n 2, respectively. Then, by definition, there exist matrices C tl), C ~2) G ~ such that n C ~1) = n a n d a C ~2) = a. S u p p o s e (i) (ii)

f o r s o m e i a n d j , ~ri -= crj, there exist kl a n d k 2 such that e i,kl ~1) = e J,k2 ~2) = 1.

T h e n the vector (x, a) is ~ - i n v a r i a n t . Proof.

We construct a matrix C such that (zc, a) C = Qr, a). Define (~(1)

C = \D~2 )

D~I)), (7c2)

235

ERGODIC TRANSFORMATIONS

where

C(rl~ ~- C(rl~ '

V (F, S) :~ (l, kl) ,

(1) = 0 , i,kl

6c2) ~2) r . s ~ t~ r,s

ej~), k 2

V (r, s) 4= (j, k2),

,

= O,

d~]~ = O,

V (r, s) :/: (i, k2),

d .i,k2 ) = 1, d~2~ -- O, d ~2~ = J,kt

V (r, s) 4= (j, kl),

1.

Let (Tr,a ) C = (p,g), where p and /a are of length n, and n2, respectively. Then tl]

n2 I L r i . r, $

,

r=l

,

r=l nl

E

_I L r L~(1) r, $

~

7~S,

if

s :/: k l ,

if

s--k

r=l

,

r¢i

_~.)+Gj r4-i --Y-°")+~,+~s, l~rCr, S

=

ttrbr,

S

I.

Similarly g~ = a~ for 1 ~ s ~< n z. Therefore, (p, #) = (n, a). We now show that A = g(C) is strongly connected. Label the vertices of g(C u)) by al,..., a,, and those of g ( C ~z)) by b I .... , b,2 in the order specified by the adjacency matrix. Then the set of vertices of A is {ak} U {bk}. The proof has three main steps. (i) V 1 ~< r ~ nl, there exists a path a , - ~ a i, and V 1 ~ r~< n2, there exists a path b r ~ bj. To see this, let a r = ar, , a~2,..., a~k = ai, be any walk in g(C°)). Let ars be the first occurrence of a i in the sequence. Then a,, ..... a,s is a path ar -* ai in g(C u)) such that every edge is in A. Hence it is a path in A. The second part of the statement is proved in a similar way. (ii) V 1 ~ r, s ~ n l, there exists a path a , ~ a~. To prove this, let a r = a~, a,: ..... a~ = a s be a path in g ( C m ) . Recall that g ( C (1)) is strongly connected. If art =/=a~ V 1 < t < k, this is also a path in A and we are done. If a~, = a~, then a~,+~ = ak, since (a~, ak, ) is the only edge leaving a~ in g ( C m ) . (Note that i4= k I and j4= k2, since otherwise g(C m ) and g(C ~2)) would not be strongly connected.) Now (ai, bkO and (bj, ak, ) are edges in A. Let

236

FRIEDMAN AND BOYARSKY

bk2---~ hj be a path, as constructed in (i). Then replace ai, akj in the path of g(C (1)) by a i, bk2~ bj, akl. The resulting path a r ~ a~ is in A. (ii') Y 1 ~< r, s <~ n 2, there exists a path b, ~ b s. The proof is the same as that in (ii). (iii) V l <~r <~ nl, ¥ l <~ s <~ n2, there exists a path a r ~ b~ and a path b~--* a r. The path ar--* b~ is a concatenation of two paths: a r --+ a i ~ bk2 -+ b s .

The path b S-~ a r is also a concatenation of two paths: b s~

hj~

akl-4 ar ,

Thus, A is strongly connected. It remains only to show that A possesses an aperiodic vertex. If either C (1) or C (2) (or both) has non-zero entries on their respective diagonals, then since i :/: k 1 and j 4= k2, C has a non-zero entry on its diagonal and therefore A has an aperiodic vertex. Finally, we show that A has an aperiodic vertex even if C (1) and C (z) have no non-zero terms on their diagonals. Note that tr tl) differs from C ~1) only in 1 term; the (i, k 0 t h entry, equal to 1, has been replaced by 0. Let Qt") denote the upper left hand block (nl × n 1 matrix) of the matrix C multiplied by itself n times. The removal of the (i, kl)th term in ~(1) causes Qtm, for N sufficiently large, to have non-zero entries in all positions except possibly in the ith row and j t h columns, since C t~) is primitive. Hence Q(N) has a non~(N) on its diagonal. Clearly qkk zero entry, say ~kk ~(N+ 1) > 0 also. Thus the vertex k is aperiodic. Hence A has an aperiodic vertex. Q.E.D. COROLLARY 4.1. Let Suppose there exists i,

m l l n I and

mzln2,

where m l , r n 2 , n ~ , n 2 ~ N .

max(m1, m2) ~< i ~< min(n 1, n2) and such that i ~ 0 m o d ml and i ~ 0 mod m2. Then = (ml, m 1 + 1,..., n 1, m 2, mE + 1,..., n2) is ~-invariant. P r o o f Follows directly from Theorem 4.1 and the construction described in Theorem 3.1. Q.E.D.

ERGODIC TRANSFORMATIONS EXAMPLE 4.1.

237

Let n = (1, 2, 3, 4, 5) and a = (3, 4, 5, 6), where

(i 1000)(111) C (1)=

0

1

0

0

0 0

0 0

1 0

0 1

1

1

l

1

3

~

5

C(2)=

and

3

0

3

3

5

0

0

1

0

0

0

0

1

l ~

1 ~

0

0

Then, by Theorem 4.1, 0z, a) C = Oz, a), where

Cz

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

1

1

1

1

0

0

0

0

0 1

0 0

1

3 0

1

31 0

~

~

5

~

0 0

0 0

0 0

0 0

3 0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

21

21

0

0

THEOREM 4.2. Let ~ and a be ~-invariant vectors of lengths nj and n 2, respectively, such that n C ( 1 ) = n and aC(2)= a. Suppose the following two conditions hoM f o r some i and j:

Cl]~i : l/x, e ~ = l/y,

(i)

n i = kx, aj = ky, where k E N.

(ii)

Then (Tr, a) is ~-invariant. Proof

Define the matrix C as follows:

C=

{ ~(1) \D(2 )

D(1) ), ((z)

where

~(1)

F,$ ~

~(1)

Urns ~

V r =2(=[,

~-(1)

1 - - , x+y

nl-x+l<~s
0,

otherwise,

Ui, S

238

F R I E D M A N AND BOYARSKY

1

do) = O, r,s

((2) r,s

~

g r 4: i,

d! ~) = -~1,$

_(2) w r 4:j, Lr,s ~ v

;(2) c j, S

l~s~y,

x+y O,

otherwise,

1 x+y

l<~s<~y,

O,

otherwise, 1

" ' r , s = O, d(2)

V r

d_.j,s ! 2) =

:# j,

n l-x+

x+y

l ~ s < < , n 1,

otherwise.

,

That (n, a ) C = (n, a) follows from the fact that 7~i

x

7~i - - +

-

x+y

Gj - -

x+y

-k

and

oj n~ aj - - - + - - - k . y x+y x+y N o w label the vertices of A = g(C) by a 1.... , a n and b~,..., bn as in the proof of Theorem4.1. Since g ( C ( 1 ) ) = g ( C °)) and g(C(2))=g(C~2)), to establish strong connectivity, it suffices to show that for all r, 1 ~< r~< n 1, and for all s, l ~ s ~ n 2, there are paths ar--+b s and b s - + a r. This is straightforward. Finally, let Q~n) represent the upper left n I × n, block of A n. Since g(~O)) = g ( C ( 1 ) ) , it is clear that Q(") >/(Ctl)) ", and hence A is aperiodic. Q.E.D.

(01000)

EXAMPLE

C ~1) =

Let n = (I, 2, 3, 4, 5) and a = (1, 2, 3, 4). Then,

4.2.

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

1

1

1

1

1

and

C (2) =

0

1 0 0)

0

0

1

0

0

0

0

1

1

1

1

1

Note that Cs,stl)= ~l and ct2)4,1= ~-L Condition (ii) is satisfied for k = 1, and the matrix C is

ERGODIC

239

TRANSFORMATIONS

1

0

0

0

0

0

0

0 0 0

1 0 0

0 0 0 1 0 0 0 1 0

0 0 0

0 0 0

0 0 0

1

1 9

1 9

1 9

_I 9

1 9

1 9

1 9

0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

1 0 0 0 1 0 0 0 1

1

1

1 ~

1 9

1 9

1 9

1 9

C=

0

0

1 9

1 9

This example generalizes to any two vectors 7¢ 1) and ~z~2), where the nith entry of 7ru) is n i and the n;th row of C Ci) consists of equal entries 1/n~, i = 1, 2. It therefore applies to all vectors of Section 2.3. EXAMPLE

4.3.

Let 7r = (2, 3, 4) and ~r = (3, 4, 5, 6), where

Oll C (1)=

1

( O, 01 1 ) ~

and

C(2)=

1

)

O0 O0 O1 O1

0

1

1

0

"

0

Then, by Theorem 4.2, (r:, tr)C = (~, o-), where 0 0

1

2 0

1

C--

0 0 0 0

0 0 0 0

z1

0

0

0

1 3

1 3

1 3

0

0

0 0 0 0

0 0 0 0

0 0 0 ~1 31 ~1 0 1 0 0 0 1

1

3

0

l

3

1

3

0

-

0

Herex=1, y=2, i=2, j=4, k=3. 4.2. Permutation Results Many of the matrices we have constructed in the foregoing sections are not in 9 , i.e., they are not permutation invariant. It is of interest, however, to investigate the extent to which these vectors can be permuted. Let C C ~ and let P be a permutation matrix. The PC is the matrix C with its rows rearranged, and the contiguity property is not affected. PCP r is the matrix which corresponds to a relabelling of the vertices of g(C). Premultiplication by pr, however, rearranges the columns of C and this may

240

FRIEDMAN AND BOYARSKY

destroy contiguity. The following two results show that certain permutations are admissible. PROPOSmON 4.1.

n=(nl

..... n , )

is ~ - i n v a r i a n t

(Tgn, 7~n_ 1 ,.-., 7"£1) is.

Proof

Let P be the skew-diagonal matrix

/

/

/

/

1)

if and

only

if n'=

1

\1 Let n be qY-invariant with C the matrix induced by n, i.e., nC = 7r. Then (PC) p r just reverses the order of the columns of PC. Thus contiguity is not affected and P C P r C ~ . Noting that n' = n P r, we have n ' P C P r = n P r P C P r = n C P r = n P r = n'.

Hence n' is c~-invariant. Similarly, we can show that n is ~-invariant if n' is, Q.E.D, PROPOSITION 4.2. S u p p o s e n C = n, where C E ~ is an n × n m a t r i x that can be written in block f o r m as (C 1, C 2, C3), a n d C i is an n × Pi m a t r i x with O ~ p j ~ n, i = 1, 2, 3. L e t C 2 have the p r o p e r t y that all its rows contain either 1 or P2 non-zero elements. Then, ~t ~ (~1 .... ' 7rpl' 0"1 ..... aP2' ~PI+P2+ 1 ' " " ~ n )

is ~ - i n v a r i a n t where ( a l ,

{7 2 , . . . ,

[702) is a p e r m u t a t i o n o f (nm + 1..... ~p2)'

Let Pz be a Pz × P 2 (rip,+ 1..... np~) P2r = (o I ..... %2)' Define

permutation

Proof

matrix

such

that

o o) P=

P2

0

0

13

,

where I i is the Pi × P i identity matrix, i = 1, 3. Then P C P r C c~, :gpr .= 7r', and n r P C P r = ~ z p r p c P r = 7cCP r = zrP r = 7r'. Q.E.D.

241

ERGODIC TRANSFORMATIONS

5. A D E N S E N E S S T H E O R E M AND A P P L I C A T I O N TO F U N C T I O N A L EQUATIONS

5.1. Denseness

Let ~ , , denote the class of non-negative, continuous, piecewise monotonic functions f ( x ) from an interval J into itself satisfying: (i) [ f ' ( x ) l ~ m , number, and

where the slope exists and m is a positive real

(ii) f ( x ) has the value 0 at all the relative minima points, i.e., all the valleys reach down to the x-axis. In this section, we shall show that a class of piecewise constant functions ~Tf° , dense (sup norm) in ~W~, is ~-invariant. Thus, given any f ~ dWm and > 0, we can construct a , C ~ such that its unique invariant density, ~, satisfies supxcs I f ( x ) -- ~(x)l < e. THEOREM 5.1. L e t ~ = ( ~ ( 1 ) , g ( 1 ) , 7 [ ( 2 ) , / . / ( 2 ) ..... ~(k),]./(k)), k / > l , be a vector such that zc") is a vector o f length ni satisfying 7r (° i-1 --p = Z j= 1 ns + P, 1 ~ p <~ n~, and Ix") is a vector o f length m i with all entries the same and • (i) equal to ~p = ~ j = il nj + I, 1 <~p <~ m i. Then ~z is c~-invariant. Before proving the theorem, we give two examples of ~: EXAMPLE 5.1. ~ = (1, 2, 3, 4, 4, 4, 5, 6, 7, 7, 8, 9, 9, 9, 9). Here 717(1) = (1, 2, 3), ~/(1) = (4, 4), zc(2) = (4, 5, 6), ///(2) = (7), 7[(3) = (7, 8), a (J) = (9, 9, 9, 9). EXAMPLE 5.2. n = (1 2 3 4 5). Here n (1) = (1 2 3 4) a n d / t (~) = (5). We remark that the last segment of n is always considered to be a separate flat segment, even if it contains only 1 entry. Also, we note that there are no restrictions on ng or m i. This feature allows us to approximate any continuous monotonic function starting at 0 and with slopes bounded by 0 and 1 by a vector of the form n thought of as a step function whose value on the ith interval is given by 7rt. Proof.

(1)

We begin by constructing the matrix C. Let i--I

bi=0;

b i = Z nj,

l
b=bk+m,

l
d = d k + x.

j=l i-1

dl=0;

d i = ~. m~, j

i

242

FRIEDMAN AND BOYARSKY

Let n = b + d. Define the following matrices for 1 ~ i <~ k: p(i) is an n i × n matrix with entries p,)=

ll,

r,s

O,

if s = b ~ + d + r , otherwise.

T (i) is an m~ × n matrix with entries

l 1 -(i) [r,s

if

bi+ 1 + 1 '

di + r <~ s <~ bi + l + di + r,

otherwise.

O,

Then we define C to be the n × n matrix IP(X)I T (1)

C=

i

•

~P(~)I \

T (k) /

(2) We show that ~zC = ~z by induction on d. For d = 1, ~z = (1, 2,..., n) and C is the same matrix that is given by the construction of Section 2. N o w suppose the result holds for vectors with d - - 1 repetitions and let 7r be a vector with d > 1 repetitions. We define a vector ff with d - 1 repetitions by deleting the first element of/~(1). That is, 7~ = (7~ (1),/.~(1) 7~(2)/./(2) ..... 7r(k) ~./(k)),

where /Y(J) is a constant vector of length m 1 - 1 (possibly 0) with entries equal to those of g(1). The construction of part (1), applied to if, yields an (n - 1) × (n -- 1) matrix ~(1)

where, for all 1 ~< i ~< k,

(,),,

,s = P r , s : ,

l <~ r<<, n i and l ~ s ~ ( n - - 1 ) ,

ERGODIC

243

TRANSFORMATIONS

and for all 2 ~< i ~ k, ~(i) r,s -- --

t(i) ~r,s+ 1'

1

<~ r ~ mi and 1 ~< s ~< (n - 1),

and ~-(1) = t(1) [r,s ~r+l,s+l,

1 <~ r ~< m 1

1 and 1 ~< s ~< (n -- 1),

i.e., C equals C with row n 1 + 1 and column 1 deleted. (We observe that, since d > 1, removal of the first repeated entry cannot cause p~k) to become O, and so ~ is of the form defined in the statement of the theorem. Also, if m 1 = l , causing tU(1) t o be empty, ~z~) and n <2) join to become a single segment of Y. However, C can still be viewed as being of the form given above with T ~1) absent. In this case, the top block of (7 given by the construction is just r~c,)~ We note that for l ~ r < , n l , c~.~=?~.s_ 1 and for n ~ + 2 < ~ r < , n , c~,~: ?~_ ~,~_ ~. By the induction hypothesis, n--1

YrC~,s = £~. r=l

Moreover, from the construction of if, z?~ = ns, 1 <~s <~ n, and 7rs_ l = n~, n ~ + 2 ~ < s ~ < n . Also, note that ~r-~_l = n s - 1 , 2 ~ < s ~ < n ~ + 1. Now, tl I 7[rCr,s =

GS~ r= 1

V ~" r=l

n

A.., ~ ~rCr,s~-7[nl+lCnl+l,s~r= 1

yr~ r , s - ~ +

1 _ _ =7~s 1-- n l + l

71) 1,s

~

~ 7-[ r - l C r- - l , s r=nt+2

+

1

t(1)

l,s"

For s ~ n 1 + 2 , -l,~t")=0 and n~_ 1" =n~. r?s_ 1 = n ~ - 1. In either case, a S = n~. (3)

TErCr,s Z r=nl + 2

For s~
t
It remains only to show that C is primitive. Since 1

Cn, n = t(k) = -b -+ >1 mk,n

O,

g(C) has an aperiodic vertex. Hence, it suffices to show that g(C) is strongly connected. We begin by partitioning the vertices {1, 2,..., n} of g(C) into 2k sets: A i={b i+d i+r:l~r~ni}, B i = {bi+ l + di + r: l ~ r <~ mi} ,

607/45/3-3

244 1 ~ i ~< k.

FRIEDMAN AND BOYARSKY

Note

that

Cbi+di+r,s = P_(1) r,s,

1 ~ r <~ n i ,

and

"(i) Cbi+l+di+r, s : lr,s,

(a) We claim there is a path from vertex n = b k + l + d k + , ~ B k to every other vertex. To prove this, we make the following two observations: (i) For 2 < ~ r ~ m r , there is an edge ( b ~ + ~ + d i + r , b i + l + d i + 1), since t r,bi+ldi+r_l (i) :~= O. (ii) For 2 < 1 < k, there is an edge (br+ L+ dr + 1, br + d~), since tli)s¢O, for d i + l ~ s < ~ b i + l + d r + 1 and d i + l < ~ b i + d i < b i + ~ + d r + 1. Therefore, for 1 ~ i ~ < k , there is a path from n to any vertex v E B ~ . Moreover, for 1 <~ i ~ k, there is an edge from a vertex in B r to all vertices in Ar. To see this, note that: r-

(iii) For l ~ l ~ < k , l~r~n r, there is an edge ( b r + l + d i + l , -I- d i + r), since d i + 1 <, b r + d i + r < bi+ ~ + d i + 1. It remains to show that there is a path from any vertex, v, to n. We do this by a backwards induction on i. br

(b) We claim there is a path from any V E A k ~ J B k to n E B k. We prove this in three steps. (iv) Suppose v = b k + d k + n k = b k + ~ + d k E A k. Then there is an edge (v, n), since n = bk+ ~ + dk+ 1. (v) If v C B k , there is a path from v to w =bk+~ + d k + 1 by step (i). By step (iii), there is an edge from w to b k + d k + nk, and, by step (iv) from there to n. (vi) If v 4 : b k + d ~ + r E A k, there is an edge (v,b k + d + r ) . If b k + d + r ~ B k, we are done. Otherwise, b k + d + r E A k, since d > dk, and we can continue constructing a strictly increasing sequence of vertices until we eventually reach a vertex in B k. (c) N o w suppose that there is a path from any v E Ap U B p to n for p > i. We claim there is a path from any v C A i U B i to n, where 1 ~< i ~< k - 1. The proof is analogous to that given above. (vii) Suppose v = b i + d i + n i = hi+ ~ + dr C A i. Then there is an edge from v to w = b i + l + d . Since d > d i + ~ ( i < ~ k - 1 ) , w E A p U B p for some p > i and we are done by the induction hypothesis. (viii) If v E B i we can show, as in step (v), that there is a path to b~+ ~ + d i and by step (vii) from there to n. (ix) Finally, if v E A r, there is a path consisting of an increasing sequence of vertices which eventually leads to Bi or to some A o or Bp, p > i. In either case, we are done. Q.E.D. We now apply the construction of the theorem to Examples 5.1 and 5.2.

ERGODIC TRANSFORMATIONS EXAMPLE 5.3.

245

d = 7, and

0

0

0

0

0

0

0

0

0

l

1

1

!

_1 !

4

4

0

0

0

0

1

0

0

0

0

0

0

0

TM

1 1 l

4

!

4

1 1

C= 1

T

1

3

_1

7

!

7

1

3

_1

7

1

1 1

!

9

_1 9 1 9

_1 9 _1

1 9 l

1 9 1

1 9 1

I 9 1

1 9 1

1 1

1

9 !

~ i

~ _1

~ _1

~ 1

g !

g _1

g !

9

9 ~.l

9 _1

9 1

9 1

9 1

9 1

9 1

9

EXAMPLE 5.4.

d = 1 and

C=

i

0'

9

g

g

g

~

~

! 9 1

~

1

~

01 0 0 \/

°0 0 0 ,

o/

0

0

0

1]

I

I

I

I]

|

Let ff be a vector as defined in the statement of Theorem 5.1, and let C c~ be the matrix constructed for it. Let # be the vector ~ with the order of the entries completely reversed, i.e., the first entry o f p is the last entry of n, etc. Thus/1 is ~-invariant and its induced matrix is C" = p ~ , p r , where P is the n X n skew-diagonal matrix. We note that the last row of C" consists of b + 1 consecutive entries, all equal to 1/(b + I), terminating in the column n. Hence the first row of C' consists of the same b + 1 consecutive entries, but now starting in column i. In view of Theorem 4.2, the vector n - - ( i f , # ) is c~-invariant. The nth and (n + 1)th rows of C are such that n C = n each consists of 2(b + 1) consecutive entries all equal to 1/2(b + 1), starting in the ( n - b - 1)th column and ending in the (n + b + 1) column. This result generalizes immediately.

246

FRIEDMAN AND BOYARSKY

THEOREM 5.2. Let 7r(1), i---- 1,..., n, be vectors of the form stated in Theorem 5.1, and let/u (~) be obtained from ~(i) by reversing the order of the entries. Let ~8"~° denote all vectors of the form (7[(1),~/./(1) ~(2) fl(2) ...,

Then every ~zE ~ ]

7~(n),[.l(n)).

is ~-invariant.

Recall, from the definition of ~?~m, that ~ is the class of non-negative, continuous, piecewise monotonic functions f from an interval J into J satisfying: (i) Idf/dx] ~ 1, where the derivative exists and (ii) the value of f ( x ) at all its relative minima is 0, i.e., each function f E ~ is a multihumped function such that each hump starts at height 0, ends at height 0, and has slope less than or equal to 1 in absolute value. The peaks of the humps are arbitrary. From Theorem 5.2, it follows that every f ~ ~ can be approximated as closely as desired (by choosing a sufficiently fine partition of J) by a step function 7r ~ ~/,0. We note that as the partition becomes finer, the resulting step functions are scaled down. Thus, we have: THEOREM 5.3.

~7"~° is dense in ~ o with respect to the uniform norm on J.

Let ~?~mhe as defined at the beginning of this section where m is a positive, fixed real number, and let ~ be any vector of the form stated in Theorem 5.2. Since 7r is cC-invariant, then clearly mT~ is cC-invariant, where the associated matrix for mTr is the same as that for 7~. Let ~ f o {mTr: 7~C~,~0}. Then, by choosing a sufficiently fine partition, we can approximate any f ~ ' ~ m uniformly as closely as desired by a piecewise constant function from ~ f °m. Thus, we have: THEOREM 5.4.

~ ° m is dense in J['~,, with respect to the uniform norm on

J. Although ~ provides a large class of ergodic density functions it is somewhat restricted by the requirement that the valleys go down to 0. We shall now briefly describe a class of multi-humped, step functions without this restriction, which can be attained exactly and whose associated transformations can be constructed. The construction and proof are straightforward, but tediousl and hence will be deleted. Let G1 denote the class of step functions zr 7~ --~ (~0), 7~(2),..., 7t(k)), where the ith segment n (;) is either fiat (i.e., it has at least two consecutive

ERGODIC

TRANSFORMATIONS

247

entries all equal to the positive, rational number p;), or it is increasing (7r~+j = ~i + 1) or it is decreasing 0zi+l = ~ i - 1). If 7t(~) is a flat segment of the function ~, let l~ denote the number of entries in ~z°) and Pi the value of these entries. THEOREM 5.5. L e t ~ C G 1. I f each fiat segment has the property that Ii >~Pi, then 7r is ~-invariant. (There are no restrictions on the length o f the sloping segments.) Remark. A sloping segment can be allowed to rise in steps o f j i C ~, rather than in steps of l, if the starting and endpoints of the sloping segment, m i and ni, satisfy mi[ n i and n i = m i d- k i j i, o r ni] m i and m i = n i + k i J i , depending on whether the segment is increasing or decreasing. As well, a flat segment need not have opposite sloping segments on either side of it, for example, a segment m a y rise in steps of 2, flatten out, then rise in steps of 1. 5.2. Functional Equations A transformation r: [0, 1] ~ R is called piecewise C 2, if there exists a partition 0 = ao < at < --' < ap = 1 of the unit interval such that for each integer i, i = 1..... p, the restriction ri of r to the open interval (ai_ ,, ai) is a C 2 function which can be extended to the closed interval [ai_l, ai] as a C 2 function. If r is non-singular, the Frobenius-Perron operator P T : - ~ --+ - ~ takes on the form P

PTU(x) = ~ f ( ~ i ( x ) ) Oi(x) Zi(X), i=1

where ~ , i = r 7 a, o i ( x ) = ]qJ/(x)] and X~ is the characteristic function of the interval J~ = r,([a,_,, a,]). We recall [7] that P ~ f = f i f and only if d~ = f d m is invariant under r. Let T = { r C { : P ¢ = = T t , = ~ g ~ ° } . Recall from Section l.2 P , , when restricted to r C {, reduces to a matrix operation, namely, PC= = ~C. With the aid of Theorem 5.4, we have: THEOREM 5.6. f

The set of f i x e d points o f the functional equations P ¢ f = ~ f(tff,(x)) o,(x) q:i(x) = f ( x ) : r E i=1

contains a set of piecewise constant functions which is dense (uniform norm on J) in the class of multi-humped functions ~T"m . Proof.

~ ~ T.

Q.E.D.

248

FRIEDMAN AND BOYARSKY 6. RANDOMIZATION AND AN EXAMPLE

6.1. Randomization

In [2], it is shown that for r C ~, the rationals are eventually periodic, i.e., for any rational number x in J there exists an integer n such that rn(x) is periodic. But the rationals are the only numbers with which computations can be performed. Thus, if n is the unique density invariant under r, it is impossible to exhibit this density by observing the iterates {za(x)}, x rational. To evade cyclic orbits in the difference equation (1.1), we randomize it as follows: for e > 0 and small, we consider the stochastic difference equation Xn+ 1 = "L'(Xn) ~- We,

(6.1)

where W, is a random variable whose density f~(x) has support on a small set containing 0. Equation (6.1) defines a stationary Markov process, which possesses the invariant density function z~,. Neglecting some technical details, we state the following result [3]:

THEOREM 6.1. L e t f ~ 60, where 6 o is the point measure at 0, and =~ denotes w e a k convergence. L e t {x~} denote the solution o f (6.1). Then 1

lim-

n~c~ n

n--I

f"

C

S~ )Cn(x~)=lrc,(x)dx-~lzc(x)dx

i=0

as e ~ O f o r all starting points x o = x o in J.

The theorem states that randomizing (1.1) creates a system whose orbit unfailingly (for all starting points) attains the invariant density zc of (1.1) as closely as desired. Thus, for example, we choose W = W, to be a uniform random variable whose support is small and centered at 0. With this fixed random variable, which is readily implemented on the computer, we can attain any qY-invariant vector n by means of (6.1). 6.2. A n E x a m p l e Let ~1 = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) and ~2 = (10, 9, 8, 7, 6, 5). Then by the construction of Section 2 and Theorem 3.1, we can construct matrices C ~1) and C (2) E ~ , where ~I i) = ~zi, i = 1, 2, and

ERGODIC

r

C (1) =_

0

1

0 0

249

TRANSFORMATIONS

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

1 10

1 10

1 10

1 10

1 10

1 10

1 10

1 10

1 10

1 10

000011TM

C (2)

Using Theorem 4.1, we get 0

1

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

1

1

1

1

1

0

:

nC = n,

3

3

s

~

where n = (n I , n2) and C E ~ is given by

0 1 1 1 1 1 1 1

C_-

0 1 10

1 10

1 10

1 10

1 10

1 10

1 10

1 10

1 10

1

1 10

1

0

0

0

0

~1

1

0

0

0

0

0

0

0

1 1 1

1 3

1 3-

1 5

1 3

1 ~

_~

F r o m C we construct the piecewise linear transformation r shown in Fig. 6.

250

FRIEDMAN AND BOYARSKY

/

16 15 14

12

/

ii

/

Y

13

i0 9 8 7 6 5 4 3 2 i !

I

I

!

I

I

I

I

I

1

2

3

4

5

6

7

8

9

! lO

lll

I 12

i 13

i 14

I 15

I 16

FIGURE 6

Let W e be a uniformly distributed random variable with mean 0 and support equal to e. Consider the stochastic difference equation

x.+ 1 = r(x.) + W.,

(6.2)

where r is given by Fig. 6 and e is small. Using (6.2), we computed the quantity

vN(i)

1

N

l

x,,(x.),

t/=0

the proportion of visits to the interval I i = ( i - 1 , i), i = 1, 2 ..... 16, in N iterations of (6.1). For e = 10 -4, x 0 = 3.5, and N = 105, we obtained

VN= (0.98,

1.96, 2.98, 4.09, 5.17, 6.20, 7.09, 7.92, 8.87, 9.99,

9.92, 8.87, 7.5, 7.06, 6.08, 4.87). Of course, choosing e smaller and N larger would yield distributions even closer to the desired piecewise constant density function lr = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 5). For example, for x = 3.5, e = 10 -7 and N = 10 6, we obtain the vector (0.994, 2.03, 3.02, 4.03, 5.05, 6.03, 7.02, 7.97, 9.01, 9.96, 9.97, 0.01, 7.99, 7.02, 6.01, 4.92).

ERGODIC TRANSFORMATIONS

251

Even for e as large as 0.01 and N as small as 10 ~, a very reasonable approximation to ~r is obtained. 7. A SPECIAL CONSTRUCTION In this section, we present a construction for rapidly growing piecewise constant densities. THEOREM 7.1. Let k C N and let zr(i) be a vector of length n >/2 with 7r~~) = U - z for all 1 <.j <, n. Then 7r ---- (;7 (1), 7~(2),..., 7C(k))

is ~-invariant. Proof. We first give a construction of an nk × nk matrix C such that ~C = ~z. Let

C(2) , C = (C(~) 1

/ where

C (1)

is defined by l,

l

n+14j<~2n,

0,

otherwise.

ci,j=l 1,

l<~j<~n,

and for 2 < ~ i ~ m ,

0,

otherwise.

For 2 ~< t ~< k-- 1, C (t) is defined by I 1

and for 2 ~ i < ~ n -

--

tn+14j4(t+l)n,

O,

otherwise

1, 1

--

(t--1)n+

n

0,

otherwise,

l <~j4tn,

252

FRIEDMAN

AND BOYARSKY

1 :

t~,j

0, Finally,

.C (k)

l <~j <. tn,

(t-2)n+

2n '

C (t).

otherwise.

is defined b y I 1 ~(k)

kn,

otherwise

O, for l < ~ i ~ n - 1 ,

l)n + 1 4J4

(k-

n

t;i, j =

and 1 _(k) _ L n, j --

2 ) n + 1 <~j <~kn,

(k-

2n otherwise.

O,

W e n o w show that n C = n. F o r 1 ~< s ~< n, ~"

~

~rCr

'

r=l

For tn + l ~ s ~

Z

s =

1 __ l~'r~ t ' r( ,ls) ~_ ~ 2 n " - -

2n

r=2

(t + l)n, l <~ t <~ k -

7[rCr s ~ ,

r=l

z...,

_ ( t ) ~ ( t ) 7t-

TLr t ' r , s

r:l

7£~t)

~-

• - -F/

.-1

Z r=2

n

1 +

n

2

2n

_}_

_Jl'r( t + 2 ) ~t(~tr+' s2 )

~

r=l

_(t+l)~(t+l) Jtr

~r,s

2'

1

_~ 7 r ( t + l ) . - - ~ - / L

2n

--.

2t

U +'

+ (n - 2) --h- + 2-n-n +

2n

= 2 t = 7cs .

F o r (k -- 1)n + 1 ~ s ~ kn, _(k 7~rCr, s =

1)

yL 1

•

r=l

__+ /'/

2k 2 --

2

~

lLr

t-r, S

r=l

2 k- 1 ~-(n--1)

=2k-l=gs" Thus, z~C = 7~.

=1=;7 s.

2,

_JLr( t + l ) ~ O( tr+, ls)

~

2 t-1 -

- -

r=l

1 :

n-- 1 --

n

2 k- 1 + - - -2- y -

_(t+2) n

1

2n

ERGODIC TRANSFORMATIONS

253

Since c2, 2 > 0, the vertex 2 of g(C) is aperiodic. It therefore only remains to show that g(C) is strongly connected. This can be done in a direct manner, We shall, however, prove the primitivity of C in a different way. We recall that an n × n matrix is fully indecomposable if there does not exist an s× (n--s)zero submatrix for some integer s [16, p. 124, Result5.4.3]. Examination of the foregoing construction reveals that C is fully indecomposable, Hence it is primitive. Q.E.D. EXAMPLE 7.1. ~ = (1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32). Here n = 2 and k = 6. Thus, " 0

0

2l

21

1

}

0

0

1

1

1

1

0

0

0

0

1

1

0

0

2

2

1

1

0

0

0

0

0

0

1 2

1 2

1

1

1

1

1 4

4

0

0

0

1 3

1 3

1 3

1

1

1

1

C= _1 4

1 4

1 4

1 4

!

!

!

!

4

4

4

4

1

I 4

2

2

1

EXAMPLE 7.2. zr = (1, 1, 1, 2, 2, 2, 4, 4, 4, 8, 8, 8). Here n = 3 and k = 4. Thus, 0 !3

0 3~

0 3~

31 0

31 0

J1 0

1 3

1 3

1 3

0

0

0

1

1

1

0

0

0

1

1

1

1

1

1

0

0

0

I

1

1

0

0

0

C=

1

1

1

1

1

!

3

~

3

1

1

1 1

1 6

!

6

1

6

1

1

3

3

1 3

1 3

1 3

!

!

!

6

6

6

254

FRIEDMAN AND BOYARSKY REFERENCES

1_ A. BOYARSKY AND M. SCAROWSKY, A class of transformations which have unique absolutely continuous invariant measures, Trans. Amer. Math. Soc. 255 (1979), 243-262. 2. M. SCAROWSKY,A. BOYARSKY,AND H. PROPPE, Properties of piecewise linear expanding maps, J. Nonlinear Anal. 4 (1980), 109-121. 3. A. BOYARSKY, Randomness implies order, J. Math. Anal. AppL 76 (1980), 483-497. 4. F. HARARY, "Graph Theory," Addison-Wesley, Reading, Mass., 1972. 5. J. GUCKENHEIMER, G. OSTER, AND A. IPAKTCHI, The dynamics of density dependent population models, J. Math. Biol. 4 (1977), 101 147. 6. A. LASOTA, Ergodic problems in biology, Socidtd Mathematique de France, Asterisque 50 (1977), 239-250. 7. A. LASOTA AND J. A. YORKE, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. 8. T-Y. LI AND J. A. YORKE, Ergodic maps on [0, 1] and nonlinear pseudo-random number generators, Nonlinear Anal. 2 (1978), 473-482. 9. T-Y. LI AND J. A. YORKE, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc. 235 (1978), 183-192. 10. E. N. LORENZ, Deterministic nonperiodic flows, J. Atmospheric Sci. 20 (1963), 130-141_ II. R. MAY, Simple mathematical models with very complicated dynamics, Nature 261 (June 10, 1976), 459-467. 12. N. METROPOLIS, M. L. STEIN, AND P. R. STEIN, On finite limit sets for transformations of the unit interval, J. Combin. Theory 15 (1973), 24-43. 13. R. F. WILLIAMS, The structure of Lorenz attractors, preprint. 14. J. LAMPERTI, "Stochastic Processes," Springer-Verlag, New York, 1977. 15. M. MARCUS AND H. MINC, "A Survey of Matrix Theory and Matrix Inequalities," Allyn & Bacon, Boston, 1964.